Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
セミパラメトリック推論の基礎の復習
Search
Daisuke Yoneoka
November 14, 2023
Research
0
59
セミパラメトリック推論の基礎の復習
Daisuke Yoneoka
November 14, 2023
Tweet
Share
More Decks by Daisuke Yoneoka
See All by Daisuke Yoneoka
感染症の数理モデル11
kingqwert
0
38
感染症の数理セミナー_10_.pdf
kingqwert
0
54
感染症の数理モデル9
kingqwert
0
49
感染症の数理モデル8
kingqwert
0
49
感染症の数理モデル7
kingqwert
0
64
感染症の数理モデル6
kingqwert
0
74
感染症の数理モデル5
kingqwert
0
76
感染症の数理モデル4
kingqwert
0
130
感染症の数理モデル3
kingqwert
0
130
Other Decks in Research
See All in Research
Weekly AI Agents News! 12月号 プロダクト/ニュースのアーカイブ
masatoto
0
320
医療支援AI開発における臨床と情報学の連携を円滑に進めるために
moda0
0
150
The Economics of Platforms 輪読会 第1章
tomonatu8
0
140
リモートワークにおけるパッシブ疲労
matsumoto_r
PRO
6
5k
博士学位論文予備審査 / Scaling Telemetry Workloads in Cloud Applications: Techniques for Instrumentation, Storage, and Mining
yuukit
1
1.7k
Human-Informed Machine Learning Models and Interactions
hiromu1996
2
570
PostgreSQLにおける分散トレーシングの現在 - 第50回PostgreSQLアンカンファレンス
seinoyu
0
230
Satellite Sunroof: High-res Digital Surface Models and Roof Segmentation for Global Solar Mapping
satai
3
140
非ガウス性と非線形性に基づく統計的因果探索
sshimizu2006
0
550
Retrieval of Hurricane Rain Rate From SAR Images Based on Artificial Neural Network
satai
3
140
DeepSeek を利用する上でのリスクと安全性の考え方
schroneko
3
750
Poster: Feasibility of Runtime-Neutral Wasm Instrumentation for Edge-Cloud Workload Handover
chikuwait
0
350
Featured
See All Featured
Done Done
chrislema
182
16k
Thoughts on Productivity
jonyablonski
69
4.5k
The MySQL Ecosystem @ GitHub 2015
samlambert
250
12k
The World Runs on Bad Software
bkeepers
PRO
67
11k
The Cost Of JavaScript in 2023
addyosmani
47
7.3k
Scaling GitHub
holman
459
140k
Refactoring Trust on Your Teams (GOTO; Chicago 2020)
rmw
33
2.8k
Cheating the UX When There Is Nothing More to Optimize - PixelPioneers
stephaniewalter
280
13k
Side Projects
sachag
452
42k
The Power of CSS Pseudo Elements
geoffreycrofte
75
5.5k
Docker and Python
trallard
44
3.3k
Imperfection Machines: The Place of Print at Facebook
scottboms
267
13k
Transcript
ηϛύϥϝτϦοΫਪͷجૅͷ෮श Daisuke Yoneoka September 29, 2014
Notations جຊతʹ Tsiatis,2006 ʹै͏. Θ͔Μͳ͔ͬͨΒࣗͰௐͯͶ! ϕΫτϧߦྻଠࣈʹͯ͠ͳ͍͚Ͳ, ͦࣗ͜Ͱิ͍ͬͯͩ͘͞. σʔλ i.i.d Ͱ
Zi = (Zi1, . . . , Zim) ∈ Rm αϯϓϧαΠζ n ਓ. i.e., Z1, . . . , Zn φ(Z) Өڹؔ u(Zi, θ) ਪఆؔ Լ͖ࣈͷ eff (ۙ) ༗ޮ (efficient) ͱ͍͏ҙຯ
ηϛύϥϝτϦοΫਪͱʁ Zi ͷີ͕ؔηϛύϥϝτϦοΫϞσϧʹै͏ͱ S = {p(z : θ, η)|θ ∈
Θ ⊂ Rr, η ∈ H} θ ༗ݶ࣍ݩͷڵຯ͋ΔύϥϝλͰ, η ແݶ࣍ݩͷͲ͏Ͱ͍͍ύ ϥϝλ (ہ֎ (nuisance) ύϥϝʔλʔ). ηϛύϥϝτϦοΫਪ: ͜ͷͱͰ θ ͷ࠷ྑͷਪఆྔ (RAL ਪఆ ྔ) ΛͱΊΔ͜ͱ
Өڹؔ θ ͳΜͰ͍͍͔Β࠷ྑΛݟ͚ͭΔͱ͍͏ͷແཧήʔ → Ϋϥε Λݶఆͯͦ͜͠Ͱݟ͚ͭΔ! (౷ܭͰΑ͘ΔΑͶ) Өڹؔ: ਪఆྔ ˆ
θ ͷӨڹؔͱ, (Ϟʔϝϯτʹ੍͕͋Δ) √ n(ˆ θ − θ) = 1 √ n n i=1 φ(Zi, θ, η) + op(1) Λຬͨ͢ϕΫτϧؔ. ˆ θ ۙઢܗਪఆྔͱݺͼ n → ∞ ͰҰகੑ ͱۙਖ਼نੑ͕͋Δ √ n(ˆ θ − θ) → N 0, E[φ(Zi, θ, η)φ(Zi, θ, η)T ] Πϝʔδతʹ͋Δσʔλ͕ͲΕ͚ͩਪఆʹӨڹΛ༩͍͑ͯΔ͔Λ දݱͨ͠ͷ
ਪఆؔͱ M ਪఆ ਪఆํఔࣜ n i=1 u(Zi, θ) ਪఆؔ =
0 ͷղͱͯ͠ಘΒΕΔͷΛ M ਪఆྔ ͱݺͿ. Α͘ݟΔ score ؔͳΜ͔ίϨ. ͨͩ͠, E[φ(Zi, θ)] = 0 ظ 0 , E[∥φ(Zi, θ)∥2] < ∞ ࢄతͳͷൃࢄ͠ͳ͍ . ͋ͱ͏গ͚ͩ݅͋͠Δ. Ұகੑͱۙਖ਼نੑΛ࣋ͭ √ n(ˆ θ − θ) = 1 √ n n i=1 E[ ∂u(Zi, θ) ∂θ ] −1 u(Zi, θ) ͕͜͜Өڹؔʹͳ͍ͬͯΔ +op(1) → N 0, E[ ∂u(Zi, θ) ∂θ ] −1 E[u(Zi, θ)u(Zi, θ)T ] E[ ∂u(Zi, θ) ∂θ ] −T ] ͜ͷۙࢄͷਪఆྔΛαϯυΠονਪఆྔͱݺΜͩΓ͢Δ
RAL ਪఆྔ ۙઢܥਪఆྔͳΜ͔ྑͦ͞͏ʂͰ super efficiency ͷ (Hodges) ͕Δʂ Super efficiency:
ۙతʹ Cramer-Rao ͷԼݶΑΓྑ͍ͷ͕Ͱ͖ Δͷ͜ͱ ͜ͷΛղܾͨ͠ͷ͕ RAL (Regular asymptotic linear) ਪఆྔ. ͦͷਖ਼ଇ݅ۃݶ͕ LDGP (local data generating process) ʹґ ଘ͠ͳ͍͜ͱ (ৄ͘͠ Tsiatis, 2006) ηϛύϥਪ͜ͷ RAL ਪఆྔͷӨڹؔΛٻΊΔ͜ͱΛߟ͑Δ
Parametric submodel ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠ p(z; θ, η) ∈ Ssub
⊂ S Λຬͨ͢ύϥϝτϦοΫϞσϧ Ssub = {p(z; θ, γ)|θ ∈ Θ ⊂ Rr, γ ∈ Γ ⊂ Rs, s ∈ N} ΛύϥϝτϦοΫαϒϞσϧͱݺͿ.
Nuisance tangent space (ہ֎ۭؒ) ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠, ύϥϝτϦοΫαϒϞσϧ Ssub ͷہ֎ۭؒΛ
TN θ,γ (Ssub) = {BT sγ(z, θ, γ)|B ∈ Rs} ͱ͢Δ. γ p(z; θ, η) ʹରԠ͢ΔͷͰ sγ(z, θ, γ) = ∂ ∂γ log p(z; θ, γ) Ͱ ද͞ΕΔ nuisance score ؔ. ͜ͷઢܗۭؒ͜ͷ nuisance score vector ʹ ΑͬͯுΒΕ͍ͯΔ. ͜ͷͱ͖ TN θ,η (S) = Ssub TN θ,γ (Ssub) Λ S ্ͷ p(z; θ, η) ʹ͓͚Δہ֎ۭؒͱΑͿ. ͪͳΈʹ, ଆͷू ߹ʹؔͯ͠ closure ΛͱΔԋࢉࢠ. Note:͜ͷۭؒେͰޙʹ, RAL ਪఆྔͷӨڹؔ͜ͷۭؒʹަۭͨؒ͠ʹ ଐ͢Δ͜ͱ͕ॏཁʹͳͬͯ͘Δʂ
ઢܗ෦ۭؒͷࣹӨͷزԿͱϐλΰϥεͷఆཧ
RAL ਪఆྔͷӨڹؔͷॏཁͳఆཧ ηϛύϥϝτϦοΫ RAL ਪఆྔ β ͷӨڹؔ φ(Z) ҎԼͷ݅Λຬ ͢Δ.
Corollary1 E[φ(Z)sβ] = E[φ(Z)sT efficient (Z, β0, η0)] = I. ͨͩ͠, s είΞؔͰ, sT efficient ༗ޮείΞؔ Corollary2 φ(Z) ہ֎ۭؒʹަ͍ͯ͠Δ. ༗ޮӨڹ্ؔͷ 2 ͭͷ݅Λຬͨ͠, ͦͷࢄߦྻ, ޮݶքΛୡ ͦ͠Ε φeffi(Z, β0, η0) = E[seff (Z, β0, η0)sT eff (Z, β0, η0)] −1 seff (Z, β0, η0)
ηϛύϥۭؒͷఆཧ ύϥϝτϦοΫαϒϞσϧͷ߹ͷ RAL ਪఆྔͷӨڹؔͱۭؒͱͷؔ Tsiatis, 2006 ͷ Ch4.3 ͋ͨΓΛݟͯͶʂ ఆཧ
1 RAL ਪఆྔͷӨڹؔ {φ(Z) + TN θ,η (S)⊥} ͱ͍͏ۭؒʹؚ·ΕΔ. ͨͩ͠, φ(Z) ҙͷ RAL ਪఆྔͷӨڹؔͰ, TN θ,η (S)⊥ ηϛύϥϝτϦο Ϋۭؒͷަิۭؒ ఆཧ 2 ηϛύϥϝτϦοΫ༗ޮͳਪఆྔ, ͦͷӨڹ͕ؔҰҙʹ well-defined Ͱܾఆ͞ Ε,φefficient = φ(Z) − {φ(Z)|TN θ,η (S)⊥} ͷཁૉ. ͪͳΈʹ, (h|U) projection of h ∈ H(ੵΛಋೖͨ͠ώϧϕϧτۭؒ) onto the space U (ઢܗۭؒ)
GEE ʹ͍ͭͯͷ Remarks Liang-Zeger ͷ GEE ͷηϛύϥϝτϦοΫϞσϧ (੍ϞʔϝϯτϞσϧ: 1 ࣍ͱ
2 ࣍ͷϞʔϝϯτʹ੍͚ͩΛஔ͍ͨϞσϧ) ҎԼͷಛΛͭ. ہॴ (ۙ༗) ޮਪఆྔ: ࢄؔͷԾఆ͕ਖ਼͚͠Ε, ༗ޮਪఆྔ Robustness: ແݶ࣍ݩͷύϥϝʔλਪఆ͕ඞཁ͕ͩ, ࢄؔΛ misspecify ͨ͠ͱͯ͠Ұகੑͱۙਖ਼نੑอ࣋ GEE ͷຊΛಡΊΘ͔Δ͚Ͳ, Working covariance matrix Λؒҧ͑ͯ ༗ޮੑࣦΘΕΔ͕, ͦͷଞͷ·͍͠ੑ࣭ (ۙਖ਼نੑͱҰகੑ) อ࣋Ͱ͖Δͬͯ͜ͱ